"The existence of 3 means that it is possible to use difference 

 scalings of variables by incorporating 6 and thus get different 

 equations of motion, especially for small beach slopes." (p. 116) 



Other derivational methods exist (e.g.. Green and Naghdi, 1976)^^ 

 which avoid using scaling parameters, but an assumed velocity field instead. 

 For a uniform vertical velocity profile, their derived equations after 

 linearization reduce to those derived by Peregrine (1967). An alternative 

 derivation has been reported by Mass and Vastano (1978)^^ using variational 

 techniques to minimize the total energy of the system. Assumptions re- 

 garding scaling of parameters are necessary. When both large wave ampli- 

 tude and bottom variations are permitted, a total of 19 terms appear in the 

 one- dimensional motion equation. Most terms are small and should disappear 

 for practical coastal engineering applications. Engineering derivations can 

 be found in Abbott and Rodenhuis (1972)^5^ Liggett (1975)^^, Abbott (1979), 

 and others. 



d. Two-Dimensional Equations in Conservation Form . For practical en- 

 gineering applications, Abbott, Petersen and Skovgaard (1978a, b) took the 

 equations derived by Peregrine (1967) but in terms of depth-integrated flows 

 and in conservation form. This gave for disturbances of small to moderate 

 amplitude over slowly varying bathymetry 



1^ + -^ + 1^ = (144) 



dt dX dy 



at ^ 8x ^h ^ ^ 9y ^h^ ^ S*" 9x 2 ^"^^ 2./^^ ^ 3x3y3t^h^J 



d'h [— ^ (^) + ^;r?^(#) ] (1^5) 



8x29t ^ ^^^y^^ ^ 



1 



9t 9y ^h^ 9x ^h^ ^ §^ 8y T'^^ ^dt ^ 9x9y9t^h^J 



^^GREEN, A.E., and NAGHDI, P.M., "A Derivation of Equations for Water 

 Propagation in Water of Variable Depth," Journal of Fluid Mechanics, 

 Vol. 78, Pt. 2, 1976, pp. 237-246 (not in bibliography). 



^'^MASS, W.J., and VASTANO, A.C., "An Investigation of Dispersive and Nondis- 

 persive Long Wave Equations Applied to Oceans of Variable Depth," Beference 

 78-8-T, Department of Oceanography, Texas A&M University, College Station, 

 Texas, July 1978 (not in bibliography). 



S^ABBOTT, M.B., and RODENHUIS, G.S., "On the Formation and Stability of the 

 Undular Hydraulic Jump," Report Series No. 10, International Courses in 

 Hydraulic and Sanitary Engineering, Delft, Netherlands (not in bibliography) 



S^LIGGETT, J. A., "Basic Equations of Unsteady Flow," Ch. 2, Unsteady Flow in 

 Open Channel, Vol. I, K. Mahmood and V. Yevjevish, eds.. Water Resources 

 Publications, Fort Collins, Colo., 1975 (not in bibliography) . 



149 



